Question

An ellipse has eccentricity $1/2$ and one focus at the point $P(1/2,1)$. Its one directrix is the common tangent, nearer to the point $P$, to the circle $x^2+y^2=1$ and the hyperbola $x^2-y^2=1$. Find the equation of the ellipse in standard form.

Answer 1

Common tangent to $x^2+y^2=1$, $x^2-y^2=1$

$x=-1$, $x=1$

Common tangent nearer to $P(\frac{1}{2},1)$ is $x=1$

Ellipse is shifting one unit upwards in y direction

$e=\sqrt{1-\frac{b^2}{a^2}}$

$\frac{1}{2}=\sqrt{1-\frac{b^2}{a^2}}$

$\frac{b^2}{a^2}=\frac{3}{4}$

$9b^2=3a^2$

Distance between focus and directrix$=|ae-\frac{a}{e}|$

$\frac{1}{2}=|\frac{1}{2}a-2a|$

$a=\frac{1}{3}$

$b=\sqrt{\frac{1}{12}}=\frac{1}{2\sqrt{3}}$

Equation in standard form:

$=\frac{(x-0{)}^2}{a^2}+\frac{(y-1{)}^2}{b^2}=1$

$=\frac{x^2}{\left(\frac{1}{3}\right)}+\frac{(y-1{)}^2}{{\left(\frac{1}{2\sqrt{3}}\right)}^2}=1$

$=\frac{x^2}{1/9}+\frac{(y-1{)}^2}{1/12}=1$.